Abstract

Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A +
\lambda B + C$ are an important class of
Hermitian matrix polynomials
with real eigenvalues, among which the overdamped quadratics are those
with nonpositive eigenvalues.
Neither the definition of overdamped nor any of the standard
characterizations provides an efficient way to test if a given $Q$
has this property.
We show that a quadratically convergent matrix iteration based on
cyclic reduction, previously studied by Guo and Lancaster,
provides necessary and sufficient conditions for $Q$ to be overdamped.
For weakly overdamped $Q$ the iteration is shown to be generically linearly
convergent with constant at worst 1/2,
which implies that
the convergence of the iteration is reasonably fast in almost all
cases of practical interest.
We show that the matrix iteration can be implemented in such a way
that when overdamping is detected a scalar $\mu<0$ is
provided that lies in the gap between the $n$ largest and $n$
smallest eigenvalues of the $n\times n$
quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$.
Once such a $\mu$ is known, the QEP can be solved by linearizing to a
definite pencil that can be reduced using already available
Cholesky factorizations to a standard Hermitian eigenproblem.
By incorporating an initial preprocessing stage that shifts a
hyperbolic $Q$ so that it is overdamped,
we obtain an efficient algorithm that identifies and solves a hyperbolic or
overdamped QEP maintaining symmetry throughout and guaranteeing real
computed eigenvalues.